General Information

• Course: Mathematics for Business I

• Professor: Sarai Vera Rodríguez

• University: UCAM Universidad Católica San Antonio

Syllabus

• Unit 1: Vector Spaces

• Unit 2: Matrix: Matrix calculus

• Unit 3: Linear equations system

• Unit 4: Linear Applications

• Unit 5: Real Matrix Diagonalization

• Unit 6: Quadratic real forms

Assessment Overview

• Theoretical Part: 80%

  • Exams: Two written exams

    • Part 1: 30% (Midterm)

    • Part 2: 50% (Final)

• Practical Part: 20%

  • Individual assignments (one assignment per unit).

  • Deliverables must be a single PDF document, well-written and legible.

  • To pass, average grade of all assignments must be 5 or greater.

Grading Criteria

• Students pass if their weighted average is >= 5 after passing all evaluative components.

• In case of failing any part (>20% weight), it must be retaken in the same academic year.

Key Concepts in Vector Spaces

• Vector Space: An algebraic structure consisting of a set where vector addition and scalar multiplication are possible.

  • Properties of Vector Addition: 8 Properties

    1. Associativity

    2. Identity element

    3. Inverse element

    4. Commutativity

  • Properties of Scalar Multiplication: 5. Associativity 6. Identity element 7. Distributivity across vector addition 8. Distributivity across scalar addition

• Linear Combinations: A linear combination of vectors is formed by multiplying and adding them using scalars.

Matrix and Operations

• Matrix Representation: A way to organize a system of equations for solving.

• Types of Matrices: Row, column, square, diagonal, scalar, and identity matrices.

• Matrix Operations: Addition and scalar multiplication defined by element-wise operations.

Determinants

• Definition: The determinant of a square matrix provides important properties regarding the matrix.

• Properties: 1. Non-null if and only if the matrix is regular. 2. Properties regarding linear combinations and order.

Systems of Linear Equations

• Identified as consistent (has solutions) or inconsistent (no solutions).

• Consistent Determined: Unique solution.

• Consistent Undetermined: Infinite solutions.

Solving Techniques

• Cramer’s Rule: For unique solutions using determinants for matrix representation of equations.

• Gauss-Jordan Method: Transform system into a diagonal matrix to find solutions.

Rouché-Frobenius Theorem

• The system is inconsistent if the rank of the coefficient matrix and augmented matrix differ.

• A consistent system exists if their ranks are equal.